Vector function

A vector function is any function whose output is a vector.

Parametric equations are often represented as vector functions in the following manner. This example has a single input ($t$) and produces three simultaneous outputs $(x, y, z)$ all which are functions of $t$ in themselves. The sum of the three components produces the radius vector.

$\vec{F}(t)=\langle x(t),y(t),z(t)\rangle =x(t)\mathbf{\hat{i}}+y(t)\mathbf{\hat{j}}+z(t)\mathbf{\hat{k}}$

Another example would be a vector function with coordinates as inputs, in the form

$\vec{F}(x,y)=p(x,y)\mathbf{\hat{i}}+q(x,y)\mathbf{\hat{j}}$

This equation represents a two-dimensional vector field.

The derivative of a vector function is equal to $\vec{F}'(t)=\left\langle\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}\right\rangle$ The integral of a vector function is equal to $\int\vec{F}(t)dt=\left\langle\int x(t)dt,\int y(t)dt,\int z(t)dt\right\rangle$